On finding widest empty curved corridors

An α-siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C such that α is the interior angle of C. Given a set P of n points in the plane and a fixed angle α, we want to compute the widest empty α-siphon that splits P into two non-empty sets. We present an efficient O(n log n)-time algorithm for computing the widest oriented α-siphon through P such that the orientation of a half-line of C is known. We also propose an O(n log n)-time algorithm for the widest arbitrarily-oriented version and an Θ(n log n)-time algorithm for the widest arbitrarily-oriented α-siphon anchored at a given point.